Question: $ 1.\overline{4} \div 3.\overline{71} = {?} $
First convert the repeating decimals to fractions. $\begin{align*} 10x &= 14.4444...\\ x &= 1.4444...\end{align*} $ $\begin{align*} 9x &= 13 \\ x &= \dfrac{13}{9}\end{align*} $ $\begin{align*} 100y &= 371.7171...\\ y &= 3.7171...\end{align*} $ $\begin{align*} 99y &= 368 \\ y &= \dfrac{368}{99}\end{align*} $ So, the problem becomes: $ \dfrac{13}{9} \div \dfrac{368}{99} = {?} $ Dividing by a fraction is the same as multiply by the reciprocal of that fraction. $ \dfrac{13}{9} \times \dfrac{99}{368} = {?} $ $ \phantom{\dfrac{13}{9} \times \dfrac{368}{99}} = \dfrac{13 \times 99}{9 \times 368} $ $ \phantom{\dfrac{13}{9} \times \dfrac{368}{99}} = \dfrac{13 \times \cancel{99}11} {\cancel{9} \times 368} $ $ \phantom{\dfrac{13}{9} \times \dfrac{368}{99}} = \dfrac{143}{368} $